Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ i^{-3}+5 i^{7} $$

Step-by-Step Solution

Answer

The simplified complex expression is \(-4i\).

Step 1: Simplifying Exponents of i

The imaginary unit, represented as 'i', satisfies the basic identity i² = -1. Knowing this, we can find the pattern of powers of i: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\) and then it repeats every four powers. Therefore, for the powers we have: 1. \(i^{-3}\): Recalling that negative exponents represent reciprocal, \(i^{-3} = 1/i^3\). Since \(i^3 = -i\), \(1/i^3 = -1/i\). To simplify, multiply numerator and denominator by the conjugate. \( \frac{-1}{i} = \frac{-1 \cdot i}{i \cdot i} = \frac{-i}{-1} = i \). 2. \(i^7\): Since the powers of i repeat every four terms, \(i^7 = i^{4+3} = i^3 = -i\). So, \(i^{-3} = i\), and \(i^7 = -i\).

Step 2: Substitute and Simplify the Expression

Substitute the simplified powers back into the original expression: \(i^{-3} + 5i^7 = i + 5(-i)\). Now, compute: 1. There is \(1i\) and \(-5i\): \(i - 5i = (1 - 5)i = -4i\).Thus, the exact expression simplifies to \(-4i\).

Key Concepts

Powers of i

When dealing with powers of a complex number, particularly the imaginary unit denoted as 'i', it is crucial to understand its repeating pattern. The imaginary unit 'i' is defined such that its square is \(i^2 = -1\). This definition gives rise to a cyclic pattern of powers of 'i':

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

This cycle repeats every four exponents, so every subsequent power can be simplified by determining its remainder when divided by four.
For example, for \(i^7\), observe that \(7 \div 4\) leaves a remainder of 3. Thus, \(i^7 = i^3\), which is \(-i\). By identifying this regularity, powers of 'i' become far easier to handle and simplify.

Imaginary Unit

The concept of the imaginary unit dates back to when mathematicians faced the challenge of finding a number that, when squared, would result in \(-1\). The solution was a new 'number', which they called the imaginary unit, \(i\). This unit allows us to expand our understanding of numbers beyond the real number line to a new dimension in complex planes.
The imaginary unit forms the basis of complex numbers in which any complex number can be expressed as \(a + bi\), where 'a' is the real part, and 'bi' is the imaginary part. The value 'b' is real, and 'i' remains consistent with \(i^2 = -1\). Understanding the imaginary unit is fundamental in working with complex numbers because it allows for calculations and expressions that include square roots of negative numbers, something real numbers cannot accommodate.

Simplifying Expressions

Simplifying expressions involving the imaginary unit often requires combining like terms and using properties of exponents. Whenever you simplify complex number expressions, it is important to:

• Use the established pattern for powers of 'i' to replace higher powers with one of the base cycle terms.
• Combine like terms, ensuring all imaginary terms are added or subtracted separately from any real numbers.

In an expression such as \(i^{-3} + 5i^7\), the simplification process involves converting each power of \(i\) using its known cycle. For \(i^{-3}\), we understand this as \(1/i^3\), which simplifies to \(i\), and \(i^7\) reduces to \(-i\) based on its position in the cycle.
Subsequently, substituting these into the expression allows you to easily combine the terms: \(i - 5i\), simplifies to \(-4i\). Simplification in complex numbers often looks to reduce expressions to the fewest terms while adhering to the rules governing 'i' and complex arithmetic.